Improved Steiner Tree Algorithms for Bounded Treewidth

We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in $\ensuremath{\mathcal{O}(B_{\ensuremath{\textit{tw}}+2}^2 \cdot \ensuremath{\textit{tw}}\ \cdot |V|)}$ time, where $\ensuremath{\textit{tw}}$ is the graph's treewidth and the Bell numberBk is the number of partitions of a k-element set. This is a linear time algorithm for graphs with fixed treewidth and a polynomial algorithm for $\ensuremath{\textit{tw}} = \ensuremath{\mathcal{O}(\log|V|/\log\log|V|)}$. While being faster than the previously known algorithms, our thereby used coloring scheme can be extended to give new, improved algorithms for the prize-collecting Steiner tree as well as the k-cardinality tree problems.

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